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Exploring new versions of DIMTEST for use with Polytomous Data Tan Li Louis Roussos Measured Progress July 24, 2009

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Outline Introduction Dimensionality Hypothesis Test Poly-DIMTEST Methods Poly-DIMTEST without AT2 Poly-NEWDIM Simulation Study Results & Conclusions Future Work

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Introduction Dimensionality Hypothesis Test Important assumption for many IRT models Equating Scoring Scaling Calibration DIF analysis Hypothesis Test H 0 : d E = 1 vs. H 1 : d E > 1

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Introduction Poly-DIMTEST ( Nandakumar, Yu, Li, & Stout, 1998 ) Hypothesis test H 0 : vs. H 1 : Split all the test items into three subtests: AT1, AT2 and PT The test statistic: Stand Error of CCOV comes from a complicated formula

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Introduction Poly-DIMTEST Weaknesses Difficulty on finding and choosing AT2 items Not enough items left for PT

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Methods Poly-DIMTEST without AT2 Based on dichotomous version of DIMTEST without AT2 (Stout, Froelich, & Gao, 2001) Steps AT and PT 1. Split all the test items into two subtests: AT and PT 2. Fit a unidimensional nonparamatric model to the original data by kernel smoothing take the place ofAT2 3. Simulate N samples from the model to take the place of AT2 The test statistic: Stand Error of CCOV comes from the same formula provided by Nandakumar, et al. (1998)

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Methods Poly-NEWDIM Based on dichotomous version of NEWDIM (Seo & Roussos, 2009) Similar procedure with Poly-DIMTEST without AT2 The test statistic: Standard Error Standard Error comes from the Standard Deviation over the simulated samples

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Simulation Study Dichotomous items All of the parameters were randomly generated from the distributions based on real data from a large multi-year pool of 729 grade 5 math items

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Simulation Study Polytomous items All of the parameters were randomly generated from the distributions based on real data from a large multi-year pool of 729 grade 5 math items

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Simulation Study Type I Error Study Power Study 2 dimensions simple structure

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Simulation Study Factors 500 examinees and 1000 examinees 52 pts test and 32 pts test AT subtest 52pts test: 5 MC, 10 MC, 2 CR, and 5 CR items 32pts test: 3 MC, 6 MC, and 3 CR items 52 pts 32 pts Dich. 5232120322080 Poly. 0510130368

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Results Type I Error for 52 points test Sample sizePD-NO-AT2PND 500 0.02040.0350 1000 0.01750.0298 Average 0.01900.0324 Type I Error for 32 points test Sample sizePD-NO-AT2PND 500 0.00640.0292 1000 0.00580.0350 Average 0.00610.0321 Power for 52 points test Sample SizePD-NO-AT2PND 500 77.6990.58 1000 89.1096.08 Average 83.4093.33 Power for 32 points test Sample SizePD-NO-AT2PND 500 54.8972.56 1000 68.8683.00 Average 61.8877.78

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Results Type I Error – 52 pts test, 400 trials Poly-DIMTEST without AT2 D,P5 MC10 MC2 CR5CR 52,03.55.5 32,51.25305.75 12,102.250.501 0,13 01.75 Poly-NEWDIM D,P 5 MC10 MC2 CR5CR 52,05.757 32,53.751.53.7511.25 12,1020.251.50.75 0,13 2.25 500 Examinees 7 Poly-DIMTEST without AT2 D,P5 MC10 MC2 CR5CR 52,02.253 32,5110.58.75 12,102.250.2501 0,13 01 Poly-NEWDIM D,P 5 MC10 MC2 CR5CR 52,054.5 32,53.50.54.257.75 12,101.25021.75 0,13 32.25 1000 Examinees

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Results Type I Error – 32 pts test, 400 trials Poly-DIMTEST without AT2 D,P 3 MC6 MC3 CR 32,012 20,310.50.75 8,6000.25 0,8 0.25 Poly-NEWDIM D,P3 MC6 MC3 CR 32,044.75 20,352.255 8,61.750.51.25 0,8 1.75 500 Examinees 7 Poly-DIMTEST without AT2 D,P 3 MC6 MC3 CR 32,00.251.5 20,310.251.25 8,6001 0,8 0 Poly-NEWDIM D,P 3 MC6 MC3 CR 32,066 20,33.7518 8,60.50.253.75 0,8 2.25 1000 Examinees

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Results Power – 52 pts test,400 trials Poly-DIMTEST without AT2 D,P 5 MC10 MC2 CR5CR 52,06196.5 32,558.7591.562.599.75 12,105192.2558.5100 0,13 60.5100 Poly-NEWDIM D,P5 MC10 MC2 CR5CR 52,083.5100 32,573.2597.2594100 12,1063.759193.25100 0,13 91100 500 Examinees < 85 ≥85 1000 Examinees Poly-DIMTEST without AT2 D,P5 MC10 MC2 CR5CR 52,079.2599.5 32,573.2597.7585100 12,1068.259985100 0,13 82.25100 Poly-NEWDIM D,P5 MC10 MC2 CR5CR 52,093.25100 32,586.2599.599100 12,1081.2597.75 100 0,13 98.25100

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Results Power – 32 pts test, 400 trials Poly-DIMTEST without AT2 D,P 3 MC6 MC3 CR 32,019.565.75 20,31756.7592.25 8,6115389.75 0,8 89 Poly-NEWDIM D,P 3 MC6 MC3 CR 32,049.7586 20,34475.598.5 8,63666.598.5 0,8 98.25 500 Examinees < 85 ≥85 Poly-DIMTEST without AT2 D,P3 MC6 MC3 CR 32,032.582.75 20,32974.7598.75 8,625.574.7598.5 0,8 98.75 Poly-NEWDIM D,P3 MC6 MC3 CR 32,065.2595.5 20,360.588100 8,654.7583.599.75 0,8 99.75 1000 Examinees

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Conclusion Type I error study Conservative Type I error behavior Poly-NEWDIM performs closer to nominal (0.05). Power study Poly-NEWDIM has greater power than Poly- DIMTEST without AT2 Poly-NEWDIM provides adequate power for a variety of conditions.

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Future Work More examinees Dimensionality structure Item parameter simulation models Develop a method to choose AT subtest for mixed MC and CR tests Real datasets Skewed ability distributions

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Reference Nandakumar, R., Yu, F., Li, H., & Stout, W. (1998). Assessing Unidimensionality of Polytomous Data. Applied Psychological Measurement, 22, 99-115. Stout, W., Froelich, A., & Gao, F. (2001). Using Resampling Methods to Produce an Improved DIMTEST Procedure. Essays on item response theory, 357-375 Seo, M., & Roussos, L. (2009). Evaluation of DIMTEST Effect-Size Measure and Its Application. Paper presented at the annual meeting of the National Council on Measurement in Education, San Diego.

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Acknowledgement Measured Progress Department of Psychometrics Dr. Louis Roussos

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